Angular momentum

What You Need to Know

• Angular momentum is essential in both classical and quantum mechanics. In classical mechanics, all isolated systems conserve angular momentum (along with energy and linear momentum). This conservation law simplifies calculations for systems such as planetary orbits, rotations of rigid bodies, and many other dynamic systems.

• In quantum mechanics, angular momentum is similarly fundamental, especially in understanding atomic structure and other systems with rotational symmetry.

• Angular momentum in QM is represented by an operator—a vector operator, to be specific, much like the momentum operator.

• However, unlike the linear momentum operator, the three components of the angular momentum operator do not commute.

L vs p Overview

Property	Linear Momentum	Angular Momentum
Nature	Linear motion in a straight line	Rotational motion around an axis
Vector form	\(\vec{p} = m \cdot \vec{v}\)	\(\vec{L} = \vec{p}\times\vec{r}\)
Scalar form	\( |\vec{p}| = m \cdot v\)	\( |\vec{L}| = I \cdot \omega\)
Mass for point particles	\(m\)	\(I=mr^2\)
Kinetic Energy	\(\frac{p^2}{2m}\)	\(\frac{L^2}{2I}\)
QM operators	\(\hat{p_x}=-i\hbar\frac{\partial}{\partial x}\)	\({\hat{L_x} = yp_z - zp_y}\)
Conservation Principle	Law of Conservation of Linear Momentum	Law of Conservation of Angular Momentum
Conservation Condition	No net external force acting on a closed system	No net external torque acting on a closed system

Classical angular momentum

Fig. 71 Angular momemntum in classical mechanics is a vector quantity defined by the cross product of position vector and linear momentum. Direction is determined by using the thumb rule.

• In classical mechanics, the angular momentum is defined via a corss product of position and linear momentum. The cross product is convenient to write using a determinant:

\[\begin{split}{\vec{L} = \vec{r}\times\vec{p} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\ x & y & z\\ p_x & p_y & p_z\\ \end{vmatrix}= \left(yp_z - zp_y\right)\vec{i} + \left(zp_x - xp_z\right)\vec{j} + \left(xp_y - yp_x\right)\vec{k}}\end{split}\]

• where \(\vec{i}, \vec{j}\) and \(\vec{k}\) denote unit vectors along the \(x, y\) and \(z\) axes. \(p_x\) and \(L_x\) are components of linear and angular momentum respectively

The Cartesian components of angular momentum

\[{L_x = yp_z - zp_y}\]

\[{L_y = zp_x - xp_z}\]

\[{L_z = xp_y - yp_x}\]

\[{\vec{L}^2 = \vec{L}\cdot\vec{L} = L_x^2 + L_y^2 + L_z^2}\]

Spherical coordinates

• Spherical coordinates are more convenient for rotational problems. We therefore replace \((x,y,z)\) by \((r, \phi, \theta)\). For instance: when considering rotation with \(r=const\) we are able to eliminate one degree of freedom associated with radial direction.

\[\hat{L}^2= L_x^2 + L_y^2 + L_z^2=\hat{L_r}^2+\hat{L_{\phi}}^2+\hat{L_\theta}^2 = \hat{L_{\phi}}^2+\hat{L_\theta}^2\]

Fig. 72 Since we are going to work in spherical coordinate system we need to know how operators look in differnet cooridnate systems. This figure shows a rectangular volume element expressed in spherical coordinates.

Geometric relations

\[x={\color{blue} r}{\color{green}sin\theta} {\color{orange} cos\phi}\]

\[y={\color{blue} r}{\color{green} sin\theta} {\color{orange} cos\phi}\]

\[z={\color{blue} r} {\color{green} cos\theta}\]

The range of spherical variables

\[Radius:\,\,\,{\color{blue} 0<{r}<\infty}\]

\[Azimuthal\, angle:{\color{green}\,\,\,0<\phi<2\pi}\]

\[Polar\, angle:\,\,\,{\color{orange} 0<\theta<\pi}\]

Volume Element

\[ dV = {\color{blue} r^2} {\color{green}\sin \theta} \, {\color{blue} dr} \, {\color{green}d\theta} \, {\color{orange}d\phi} \]

Laplacian

Cartesian

\[ \nabla^2 = \frac{{\partial^2}}{{\partial x^2}} + \frac{{\partial^2}}{{\partial y^2}} + \frac{{\partial^2}}{{\partial z^2}} \]

Polar

\[ \nabla^2 = {\color{blue}\frac{1}{{ r^2}} \frac{{\partial}}{{\partial r}} \left(r^2 \frac{{\partial}}{{\partial r}}\right) } + \frac{1}{{{\color{blue} r^2} {\color{orange}\sin(\phi)}}} {\color{green}\frac{{\partial}}{{\partial \theta}} \left(\sin(\phi) \frac{{\partial}}{{\partial \theta}}\right)} + {\color{orange}\frac{1}{{{\color{blue} r^2} \sin^2(\phi)}} \frac{{\partial^2}}{{\partial \phi^2}}} \]

Example

Compute volume of cube and sphere using cartesian and spherical cooridnates by integrating volume elements

Solution

\[\int^a_0\int^b_0\int^c_0 dxdydz=a\cdot b\cdot c\]

\[\int^r_0\int^{2\pi}_0\int^\pi_0 r^2 \sin \theta \, dr \, d\theta \, d\phi=\frac{r^3}{3}\Big|^r_0 \cdot 2\pi \cdot (-cos\theta) \Big|^\pi_0\cdot 2\pi=\frac{4\pi r^3}{3}\]

Example

Write down laplacian for a rigid rotor problem \(r=const\). Show what equations result when you separate the two angular variables by pluggin in \(\psi(r, \theta, \phi)\)

Solution

\[ \nabla^2 = {\color{blue}\frac{1}{{ r^2}}}\Big[ {\color{green}\frac{1}{\sin(\theta)}}{\color{green}\frac{{\partial}}{{\partial \theta}} \left(\sin(\theta) \frac{{\partial}}{{\partial \theta}}\right)} + {{\color{green}\frac{1}{{ \sin^2(\theta)}}} {\color{orange} \frac{{\partial^2}}{{\partial \phi^2}}}}\Big] = {\color{blue}\frac{1}{r^2}}\nabla^2_{\theta, \phi} \]

Quantum angular momentum

• In quantum mechanics, the classical angular momentum is replaced by the corresponding quantum mechanical operator.

• In spherical coordinates, the angular momentum operators can be written in the following form. The derivations are quite tedious involving multiple applications of chain rule but its just a straightforward math procedure. Note that the choice of \(z\)-axis here was arbitrary. Sometimes the physical system implies such axis naturally (for example, the direction of an external magnetic field).

Cartesian

\[{\hat{L}_x = -i\hbar\left(y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y}\right)}\]

\[{\hat{L}_y = -i\hbar\left(z\frac{\partial}{\partial x} - x\frac{\partial}{\partial z}\right)}\]

\[{\hat{L}_z = -i\hbar\left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right)}\]

\[\hat{L}^2=\hat{L_x}^2+\hat{L_y}^2+\hat{L_z}^2\]

Polar

\[{\hat{L}_x = i\hbar\left(\sin(\phi)\frac{\partial}{\partial\theta} + \cot(\theta)\cos(\phi)\frac{\partial}{\partial\phi}\right)}\]

\[{\hat{L}_y = i\hbar\left(-\cos(\phi)\frac{\partial}{\partial\theta} + \cot(\theta)\sin(\phi)\frac{\partial}{\partial\phi}\right)}\]

\[{\hat{L}_z = -i\hbar\frac{\partial}{\partial\phi}}\]

\[{\vec{\hat{L}}^2 = -\hbar^2\underbrace{\left[\frac{1}{\sin(\theta)}\frac{\partial}{\partial\theta}\left(\sin(\theta)\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}\right]}_{\equiv \Lambda^2}}\]

Components of angular momentum do not commute!

• Unlike linear momentum components of angular momentum do not commute!

• This implies that it is not possible to measure any of the Cartesian angular momentum pairs simultaneously with an infinite precision (the Heisenberg uncertainty relation).

• Note the cyclical nature of commutations between components

Eignefunctions and eigenvalues of \(L\) and \(L_z\)

• Since operators of component and angular momentum square commute is possible to find functions that are eigenfunctions of both \(\vec{\hat{L}}^2\) and \(\hat{L}_z\).

Eignefunctions and eigenvalues of angular momentum

\[{{\hat{L}}^2 Y_l^m(\theta,\phi) = \hbar^2 l(l+1)\cdot Y_l^m(\theta,\phi)}\]

\[{\hat{L}_zY^m_l(\theta,\phi) = m\hbar \cdot Y_l^m(\theta,\phi)}\]

• Eigenfunctions: \(Y_l^m(\theta,\phi) \)

• Angular quantum number: \(l = 0,1,2,3...\)

• Magnetic quantum number: \(|m| = 0,1,2,3,...l\)

• Note that here \(m\) has nothing to do with magnetism but the name originates from the fact that (electron or nuclear) spins follow the same laws of angular momentum.

• Functions \(Y_l^m\) are called spherical harmonics. Examples of spherical harmonics with various values of \(l\) and \(m\) are given below (with Condon-Shortley phase convention

Spherical harmonics

• Spherical harmonics are important in many theoretical and practical applications, e.g., the representation of multipole electrostatic and electromagnetic fields, computation of atomic orbital electron configurations, representation of gravitational fields, MRI imaging for streamline tractography, and the magnetic fields of planetary bodies and stars.

• The nodes of spherical harmonics, \(Y_{lm}(\theta, \phi)\), are determined by the angular part of the solutions to the Laplace equation in spherical coordinates. The nodal structure is influenced by both the degree \( l \) and the order \(m\), defining the number and type of nodes as follows:

1. Colatitudinal Nodes (Polar Nodes): The variable \( \theta \) defines “polar” nodes, which appear as circular bands around the sphere. The expression in \( \cos(\theta) \) or \( \sin(\theta) \) creates colatitudinal nodes:

   • For example, \( Y_{10} \) with \( \cos(\theta) \) has a single node at \( \theta = \pi/2 \), creating an equatorial node.

   • \( Y_{20} \), with \( (3\cos^2(\theta) - 1) \), introduces two polar nodes, dividing the sphere into three regions along the colatitude.

2. Azimuthal Nodes (Longitudinal Nodes): The variable \( \phi \) defines “azimuthal” nodes due to the terms \( e^{im\phi} \), which create lines of longitude where the function changes phase. The number of azimuthal nodes is determined by \( |m| \):

   • If \( m = 0 \), there are no azimuthal nodes, as seen in \( Y_{00} \) and \( Y_{10} \).

   • For \( m = \pm 1 \), a single azimuthal node occurs (e.g., \( Y_{1-1} \), \( Y_{2-1} \)), and for \( m = \pm 2 \), two azimuthal nodes appear (e.g., \( Y_{2-2} \)).

3. Total Nodes: equal to sum of nodes coming from polar and longitudinal nodes and are exactly equal to \(l\). E. gl=0 no nodes, l=4 four nodes.

Summary of Nodes for Each \( Y_{lm} \)

\( Y_{lm}(\theta, \phi) \)	Expression	Colatitudinal Nodes	Azimuthal Nodes
\( Y_{00} \)	\( \frac{1}{\sqrt{4\pi}} \)	0	0
\( Y_{10} \)	\( \sqrt{\frac{3}{4\pi}} \cos(\theta) \)	1 (equatorial)	0
\( Y_{1-1} \)	\( \sqrt{\frac{3}{4\pi}} \sin(\theta) e^{-i\phi} \)	0	1
\( Y_{20} \)	\( \sqrt{\frac{5}{16\pi}} (3\cos^2(\theta) - 1) \)	2 (colatitudinal rings)	0
\( Y_{2-1} \)	\( \sqrt{\frac{15}{4\pi}} \sin(\theta) \cos(\theta) e^{-i\phi} \)	1 (equatorial)	1
\( Y_{2-2} \)	\( \sqrt{\frac{15}{4\pi}} \sin^2(\theta) e^{-2i\phi} \)	0	2

Orthogonality

The orthogonality of spherical harmonics is expressed as:

\[ \langle l', m'| l, m \rangle = \delta_{l,l'}\delta_{m,m'} \]

This orthogonality condition can be explicitly written as:

\[ \int_0^{\pi} \int_0^{2\pi} Y_{l'}^{m'*}(\theta, \phi) Y_l^m(\theta, \phi) \sin(\theta) \, d\theta \, d\phi = \delta_{l,l'}\delta_{m,m'} \]

• Mathematical Structure: The spherical harmonics incorporate alternating odd and even functions of associated Legendre polynomials, much like Hermite polynomials in the quantum harmonic oscillator.

• Visual Representation: Spherical harmonics exhibit distinct nodal patterns that increase with higher quantum numbers. This behavior mirrors the nodal structure seen in other quantum systems, such as the particle in a box and the harmonic oscillator.

• Symmetry and Orthogonality: By symmetry, the product of two different spherical harmonics integrated over the sphere equals zero, a property that directly reflects their orthogonality.

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import numpy as np
from scipy.special import sph_harm
from scipy.integrate import dblquad

def orthogonality_test(l1, m1, l2, m2):
    """
    Computes the orthogonality integral of spherical harmonics Y(l1, m1) and Y(l2, m2).
    
    Parameters:
    l1, m1 (int): Degree and order of the first spherical harmonic.
    l2, m2 (int): Degree and order of the second spherical harmonic.
    
    Returns:
    float: The result of the orthogonality integral.
    """
    # Define the integrand for the orthogonality condition
    def integrand(theta, phi):
        Y_lm1 = sph_harm(m1, l1, phi, theta)
        Y_lm2 = sph_harm(m2, l2, phi, theta)
        return np.real(Y_lm1 * np.conj(Y_lm2)) * np.sin(theta)
    
    # Perform the integration over theta (0 to pi) and phi (0 to 2*pi)
    integral, error = dblquad(integrand, 0, 2 * np.pi, lambda _: 0, lambda _: np.pi)
    return integral

# Test orthogonality between different spherical harmonics
print("Orthogonality Tests:")
print(f"∫ Y(1,0) * Y(1,0)* = {orthogonality_test(1, 0, 1, 0):.5f} (should be close to 1)")
print(f"∫ Y(1,0) * Y(1,1)* = {orthogonality_test(1, 0, 1, 1):.5f} (should be close to 0)")
print(f"∫ Y(1,1) * Y(2,1)* = {orthogonality_test(1, 1, 2, 1):.5f} (should be close to 0)")
print(f"∫ Y(2,1) * Y(2,1)* = {orthogonality_test(2, 1, 2, 1):.5f} (should be close to 1)")
print(f"∫ Y(2,0) * Y(2,-1)* = {orthogonality_test(2, 0, 2, -1):.5f} (should be close to 0)")

Orthogonality Tests:
∫ Y(1,0) * Y(1,0)* = 1.00000 (should be close to 1)
∫ Y(1,0) * Y(1,1)* = -0.00000 (should be close to 0)
∫ Y(1,1) * Y(2,1)* = 0.00000 (should be close to 0)
∫ Y(2,1) * Y(2,1)* = 1.00000 (should be close to 1)
∫ Y(2,0) * Y(2,-1)* = 0.00000 (should be close to 0)

Plotting Spherical Harmonics

• Here we will visualize spherical harmonic on a 3D sphere of radius 1. Radidus is fixed so there are only two variables which dictate nodal features:

• Colatitudinal nodes depend on \( l \) and represent bands around the sphere, arising from terms in \( \cos(\theta) \) or \( \sin(\theta) \).

• Azimuthal nodes depend on \( |m| \) and create longitudinal nodes based on \( e^{im\phi} \).

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import matplotlib.pyplot as plt
from matplotlib import cm
import numpy as np
from scipy.special import sph_harm
from mpl_toolkits.mplot3d import Axes3D

def generate_spherical_harmonic_data(l, m):
    """
    Generates the data needed to plot the spherical harmonic for given l and m values.
    
    Parameters:
    l (int): Degree of the spherical harmonic.
    m (int): Order of the spherical harmonic.

    Returns:
    tuple: (x, y, z, fcolors_normalized) where:
        - x, y, z are the Cartesian coordinates of the spherical surface,
        - fcolors_normalized is the color data for plotting.
    """
    
    # Define theta and phi grids
    phi = np.linspace(0, np.pi, 100)          # colatitude
    theta = np.linspace(0, 2 * np.pi, 100)    # azimuth
    phi, theta = np.meshgrid(phi, theta)

    # Cartesian coordinates for the unit sphere
    x = np.sin(phi) * np.cos(theta)
    y = np.sin(phi) * np.sin(theta)
    z = np.cos(phi)

    # Calculate the spherical harmonic Y(l, m) and normalize it to [-1, 1] for color mapping
    fcolors = sph_harm(m, l, theta, phi).real
    fcolors_normalized = (fcolors - fcolors.min()) / (fcolors.max() - fcolors.min())
    
    return x, y, z, fcolors_normalized

\(l=1\) harmonics

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# Define the range of m and l values for the first three spherical harmonics
harmonics = [(0, 1), (1, 1), (-1, 1)]  # List of (m, l) pairs to plot

# Create a figure to hold three subplots in one row
fig = plt.figure(figsize=(18, 6))
fig.suptitle("First Three Spherical Harmonics", fontsize=16)

for i, (m, l) in enumerate(harmonics, start=1):

    x, y, z, fcolors_normalized = generate_spherical_harmonic_data(l, m)

    # Create a subplot for each (m, l) pair in one row
    ax = fig.add_subplot(1, 3, i, projection='3d')
    ax.plot_surface(x, y, z, rstride=1, cstride=1, facecolors=cm.seismic(fcolors_normalized), 
                    linewidth=0, antialiased=False)

    # Customize each subplot
    ax.set_title(f"$Y_{{{l},{m}}}$", fontsize=14)
    ax.set_box_aspect([1, 1, 1])  # Ensures spherical aspect ratio
    ax.set_axis_off()             # Turn off axes for clarity

# Adjust layout for a clear view of all subplots in one row
plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.show()

\(l=2\) harmonics

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import matplotlib.pyplot as plt
from matplotlib import cm
import numpy as np
from scipy.special import sph_harm
from mpl_toolkits.mplot3d import Axes3D

# Define l and the range of m values for spherical harmonics with l=2
l = 2
m_values = [-2, -1, 0, 1, 2]

# Create a figure to hold five subplots in a 2x3 layout
fig = plt.figure(figsize=(15, 10))
fig.suptitle("Spherical Harmonics with l=2", fontsize=18)

for i, m in enumerate(m_values, start=1):

    # Calculate the spherical harmonic Y(l, m) and normalize it to [-1, 1] for color mapping
    x, y, z, fcolors_normalized = generate_spherical_harmonic_data(l, m) 

    # Create a subplot for each (m, l) pair in a 2x3 grid
    ax = fig.add_subplot(2, 3, i, projection='3d')
    ax.plot_surface(x, y, z, rstride=1, cstride=1, facecolors=cm.seismic(fcolors_normalized),
                    linewidth=0, antialiased=False)

    # Customize each subplot
    ax.set_title(f"$Y_{{2,{m}}}$", fontsize=16)
    ax.set_box_aspect([1, 1, 1])  # Ensures spherical aspect ratio
    ax.set_axis_off()             # Turn off axes for clarity

# Adjust layout for a clear view of all subplots in a 2x3 grid
plt.tight_layout(rect=[0, 0, 1, 0.95])
plt.show()

Total number of nodes = \(l\)

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import numpy as np
import plotly.graph_objects as go
from scipy.special import sph_harm

# Define the range of m and l values for the first three spherical harmonics
harmonics = [(1, 1), (1, 2), (2, 3)]  # List of (m, l) pairs to plot

# Initialize the figure
fig = go.Figure()


# Loop over each harmonic and create a separate subplot
for i, (m, l) in enumerate(harmonics, start=1):
    
    # Calculate the spherical harmonic Y(l, m) and normalize it to [0, 1]
    x, y, z, fcolors_normalized = generate_spherical_harmonic_data(l, m) 
    
    # Add the spherical harmonic to the figure as a surface in a new scene
    fig.add_trace(go.Surface(
        x=x, y=y, z=z,
        surfacecolor=fcolors_normalized,
        colorscale='rdbu',
        showscale=False,
        name=f"$Y_{{{l},{m}}}$",
        scene=f'scene{i}'  # Assign each plot to a different scene
    ))

# Update layout to arrange scenes in a row and add titles as annotations
fig.update_layout(
    title="First Three Spherical Harmonics",
    scene=dict(domain=dict(x=[0, 0.33]), xaxis=dict(visible=False), yaxis=dict(visible=False), zaxis=dict(visible=False), aspectmode='cube'),
    scene2=dict(domain=dict(x=[0.33, 0.66]), xaxis=dict(visible=False), yaxis=dict(visible=False), zaxis=dict(visible=False), aspectmode='cube'),
    scene3=dict(domain=dict(x=[0.66, 1]), xaxis=dict(visible=False), yaxis=dict(visible=False), zaxis=dict(visible=False), aspectmode='cube'),
    annotations=[
        dict(text="$Y_{2,0}$", x=0.16, y=1.05, showarrow=False, xref="paper", yref="paper", font=dict(size=14)),
        dict(text="$Y_{2,1}$", x=0.5, y=1.05, showarrow=False, xref="paper", yref="paper", font=dict(size=14)),
        dict(text="$Y_{2,-1}$", x=0.83, y=1.05, showarrow=False, xref="paper", yref="paper", font=dict(size=14))
    ]
)

# Display the figure
fig.show()

Show code cell source Hide code cell source

# Generate data for Y(11,1)
l, m = 7, 1
x, y, z, fcolors_normalized = generate_spherical_harmonic_data(l, m)


# Plot the spherical harmonic Y(11,1)
fig = go.Figure(data=[go.Surface(
    x=x, y=y, z=z,
    surfacecolor=fcolors_normalized,
    colorscale='rdbu',
    cmin=0,
    cmax=1,
    showscale=False
)])

# Customize the layout
fig.update_layout(
    title=f"Spherical Harmonic $Y_{{{l},{m}}}$",
    scene=dict(
        xaxis=dict(visible=False),
        yaxis=dict(visible=False),
        zaxis=dict(visible=False),
        aspectmode='cube'
    )
)

# Display the figure
fig.show()

Learn More about Spherical Harmonics